Problem 5 - Introductory Algebra for College Students

Question

Solve each equation using the quadratic formula. Simplify irrational solutions, if possible. $$x^{2}+4 x-6=0$$

Step-by-Step Solution

Answer

The roots of the equation are \(x =1−2\sqrt{10}\) and \(x=1+2\sqrt{10}\), which are approximately \(-0.449\) and \(-13.551\) respectively

Step 1: Identify the coefficients

In the given equation \(x^{2}+4x-6=0\), the coefficient 'a' of \(x^2\) is 1, 'b' the coefficient of 'x' is 4, and 'c' the constant term is -6.

Step 2: Substitute the coefficients into the quadratic formula

Substitute 'a', 'b' and 'c' into the formula, \(x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}\), which gives \(x = \frac{-4 ± \sqrt{4^2 - 4(1)(-6)}}{2(1)}\).

Step 3: Simplify the equation

Simplify the equation further to get \(x = \frac{-4 ± \sqrt{16 + 24}}{2} = \frac{-4 ± \sqrt{40}}{2}\).

Step 4: Calculate the roots

The roots of the equation are \(x = \frac{-4 + \sqrt{40}}{2}\) and \(x = \frac{-4 - \sqrt{40}}{2}\). In decimal format, these are approximately \(-0.449\) and \(-13.551\), respectively. Note that \(\sqrt{40}\) further simplifies to \(2\sqrt{10}\). Hence, the simplified roots actually are \(x =1−2\sqrt{10}\) and \(x=1+2\sqrt{10}\).

Key Concepts

Solving Quadratic Equations

When we're faced with a quadratic equation like \(x^2 + 4x - 6 = 0\), we have several methods to solve it, but using the quadratic formula is often the most reliable when the equation doesn't factor easily. The quadratic formula is given as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

By identifying the coefficients from the equation:\(a = 1\), \(b = 4\), and \(c = -6\), we can substitute these into the formula. The "\(b^2 - 4ac\)" part inside the square root is called the discriminant. It tells us about the nature of the roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is one real root (a repeated root).
• If \(b^2 - 4ac < 0\), there are no real roots (the roots are complex).

In our equation, \(b^2 - 4ac = 16 + 24 = 40\), which is positive, confirming there are two distinct real roots.

Simplifying Irrational Solutions

Once we've found the square root of the discriminant, in this case \(\sqrt{40}\), it's important to simplify it if possible, because it makes your answer cleaner and more precise. An irrational number cannot be expressed as a simple fraction and usually appears under a square root or involves terms like \(\pi\) or \(e\).To simplify \(\sqrt{40}\), notice that \(40 = 4 \times 10\). Therefore, \(\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}\). Thus, the simplified roots are \(x = \frac{-4 \pm 2\sqrt{10}}{2}\), and we continue simplifying to get the final roots \(x = -2 \pm \sqrt{10}\).

Algebraic Roots

The roots of a quadratic equation represent where the graph of the equation crosses the x-axis. Sometimes these roots are whole numbers, fractions, or they could be more complex involving irrational numbers or complex numbers.In our problem, after simplifying, we found the roots as \(-2 + \sqrt{10}\) and \(-2 - \sqrt{10}\). These roots are algebraic because they appear as solutions to a non-linear polynomial equation (our original quadratic equation).Algebraic roots can tell us a lot about how a function behaves, such as intervals of increase/decrease, and can also be used in further mathematical operations or modeling real-world situations.